
This is a direct consequence of Proposition REF .

The result immediately follows from the computation of \(\mbox{cov}_{\pi }(X_0(k),X_n(\ell ))\) , for any \(k,\ell \in [\![-K,K]\!]\) :
\(\mbox{cov}_{\pi }(f(X_0),f(X_n)) & = & \sum _{k,\ell =-K}^Ka_ka_\ell \mbox{cov}_\pi \big (X_0(k),X_n(\ell )\big )\\& = & \sum _{k,\ell =-K}^Ka_ka_\ell \Big (\mathbb {E}_{\pi }[X_0(k)X_n(\ell )]-\mu _k\mu _\ell \Big ).\) 
We start with the case \(k\in [\![1,K]\!]\)  and \(\ell \in [\![-K,0]\!]\) . Using representation (REF ), we see that \(X_n(\ell )\)  is a function of \((X_0(-\ell ),\ldots ,X_0(0),X_1(0),\ldots ,X_n(0))\) . Therefore, \(X_0(k)\)  and \(X_n(\ell )\)  are independent.
The case \(k\in [\![-K,0]\!]\)  and \(\ell \in [\![1,K]\!]\)  is similar. First, we observe that \(X_n(\ell )\)  is a function of \((X_0(1),\ldots ,X_0(\ell ),X_1(0),\ldots ,X_n(0))\) , and is therefore again independent of \(X_0(k)\) .
In both of these cases, \(\mbox{cov}_\pi (X_0(k),X_n(\ell ))=0\) .
Next, we look at the case \(k,\ell \in [\![1,K]\!]\) . Using representation (REF ) and the independence of \(X_0,X_1(0),X_2(0),\ldots \) , we write
\(\mathbb {E}_{\pi }[X_0(k)X_n(\ell )] & = & \mathbb {E}_{\pi }\left[X_0(k)\left(\prod _{j=0}^{\ell -1}X_0(\ell -j)^{\alpha _{j,n}}\right)\left(\prod _{m=1}^nX_{n-m+1}(0)^{\nu _{\ell ,m}}\right)\right]\\& = & \mathbb {E}_{\pi }\left[X_0(k)\left(\prod _{j=0}^{\ell -1}X_0(\ell -j)^{\alpha _{j,n}}\right)\right]\left(\prod _{m=1}^n\mathbb {E}_{\pi }[X_{n-m+1}(0)^{\nu _{\ell ,m}}]\right)\) 
which, in the case \(k<\ell \) , yields to \(\mathbb {E}_{\pi }[X_0(k)X_n(\ell )]=0\)  in view of the fact \(\alpha _{0,n}=1\)  and, for \(\ell \ge 1\) , \(X_0(\ell )\)  is symmetric. The case \(k>\ell \)  is dealt with in exactly the same way yielding again \(\mathbb {E}_{\pi }[X_0(k)X_n(\ell )]=0\) . Finally, we look at the case \(k=\ell \ge 1\) :
\(\mathbb {E}_{\pi }[X_0(k)X_n(k)] = \left(\prod _{j=1}^{k-1}\mathbb {E}_{\pi }[X_0(k-j)^{\alpha _{j,n}}]\right)\left(\prod _{m=1}^n\mathbb {E}_{\pi }[X_{n-m+1}(0)^{\nu _{k,m}}]\right).\) 
Since the \(X_0(k-j)\)  are symmetric, the first product takes the value 1 if \(\alpha _{j,n}=0\)  for any \(j\in [\![1,k-1]\!]\) , and takes the value 0 otherwise. Using Lemma REF  of the Appendix, we see that, letting \(2^\kappa \)  be the largest power of 2 that divides \(n\) , if \(k\le 2^\kappa \) , then the aforesaid product is 1, and if \(k>2^\kappa \) , then the product is 0. The second product equals \((2p-1)^{\omega _{k,n}}\) .
The last case is \(k,\ell \in [\![-K,0]\!]\) . Here, our focus is the upward process \((\widehat{X}_n)_n\) , which reaches stationarity after \(K+1\)  steps (see Proposition REF ):
\({\mathbb {E}_{\pi }[X_0(k)X_n(\ell )]\ =\ \mathbb {E}_{\widehat{\pi }}[\widehat{X}_0(k)\widehat{X}_n(\ell )]\ =\ \mathbb {E}[\widehat{X}_{K+1}(k)\widehat{X}_{n+K+1}(\ell )]}\\& = & \mathbb {E}\left[\left(\prod _{j=0}^{-k}X_{K+1-j}(0)^{\beta _{-k,j}}\right)\left(\prod _{j=0}^{-\ell }X_{n+K+1-j}(0)^{\beta _{-\ell ,j}}\right)\right]\\& = & \mathbb {E}\left[\left(\prod _{m=K+1+k}^{K+1}X_m(0)^{\beta _{-k,K+1-m}}\right)\left(\prod _{m=n+K+1+\ell }^{n+K+1}X_m(0)^{\beta _{-\ell ,n+K+1-m}}\right)\right].\) 
Clearly, in the case \(n+\ell \ge 1\) , the two products are independent and we have once more \(\mbox{cov}_\pi (X_0(k),X_n(\ell ))=0\) . If \(n\le -\ell \) , then we can rearrange the above products in the 3 independent products the means of which are: \(\prod _{j\in I_1\setminus I_2}(2p-1)^{\beta _{-k,j}}\) , \(\prod _{j\in I_1\cap I_2}(2p-1)^{\beta _{-k,j}+\beta _{-\ell ,n+j}-2\beta _{-k,j}\beta _{-\ell ,n+j}}\)  and \(\prod _{j\in I_2\setminus I_1}(2p-1)^{\beta _{-\ell ,n+j}}\) ,
where \(I_1=[\![0,-k]\!]\)  and \(I_2=[\![-n,-n-\ell ]\!]\) . However, for \(-k\le -n-\ell \) ,
\({\sum _{j\in I_1\setminus I_2}\beta _{-k,j} + \sum _{j\in I_1\cap I_2}\big (\beta _{-k,j}+\beta _{-\ell ,n+j}-2\beta _{-k,j}\beta _{-\ell ,n+j}\big ) + \sum _{j\in I_2\setminus I_1}\beta _{-\ell ,n+j}}\\& = & \sum _{j=0}^{-k}\beta _{-k,j} + \sum _{j=0}^{-k}\beta _{-\ell ,n+j} - 2\sum _{j=0}^{-k}\beta _{-k,j}\beta _{-\ell ,n+j} + \sum _{j=-n}^{-1}\beta _{-\ell ,n+j} + \sum _{j=-k+1}^{-n-\ell }\beta _{-\ell ,n+j}\\& = & 2^{\theta (-k)} + 2^{\theta (-\ell )} - 2\sum _{j=0}^{-k}\beta _{-k,j}\beta _{-\ell ,n+j}.\) 
Similarly, if \(-k>-n-\ell \) ,
\({\sum _{j\in I_1\setminus I_2}\beta _{-k,j} + \sum _{j\in I_1\cap I_2}\big (\beta _{-k,j}+\beta _{-\ell ,n+j}-2\beta _{-k,j}\beta _{-\ell ,n+j}\big ) + \sum _{j\in I_2\setminus I_1}\beta _{-\ell ,n+j}}\\& = & \sum _{j=-n-\ell +1}^{-k}\beta _{-k,j} + \sum _{j=0}^{-n-\ell }\beta _{-k,j} + \sum _{j=0}^{-n-\ell }\beta _{-\ell ,n+j} - 2\sum _{j=0}^{-n-\ell }\beta _{-k,j}\beta _{-\ell ,n+j} + \sum _{j=-n}^{-1}\beta _{-\ell ,n+j}\\& = & 2^{\theta (-k)} + 2^{\theta (-\ell )} - 2\sum _{j=0}^{-n-\ell }\beta _{-k,j}\beta _{-\ell ,n+j}.\) 
We immediately get that
\(\mathbb {E}_{\pi }[X_0(k)X_n(\ell )] = (2p-1)^{\mathcal {B}(-k,-\ell ,n)}.\) 

Theorem 17.0.1 of [1]} states that
\(\sigma ^2 = \mbox{var}_\pi (f(X_0))+2\sum _{n=1}^\infty \mbox{cov}_{\pi }(f(X_0),f(X_n)).\) 
Substituting the expressions from 1. and 2. leads to the required representation.

Finally, the positivity of \(\Sigma (k,k)\)  is a direct consequence of Proposition REF . Simply take \(a_k=1\)  and \(a_\ell =0\)  for \(\ell \ne k\) .

